1DFEM Implementation

0)1()0(

')''(

uu

fcubuau

2)( xxa

0)( xb1)( xc

33)( 2 xxxf

Example:

MM

MM

ff

ff

ff

ff

hb

1

12

32

21

2

4100

1

00

41

0014

6

hM

121

1

211

110

00

00

00

1K

MMM

M

aaa

a

aaa

aaa

h

% example_1D.m filesclearn = 7; % no interior pointsh = 1/(n+1);a = inline('x+2','x');c = inline('1','x');f = inline('-x^2+5*x+3','x');uexact = inline('x*(1-x)','x'); K = sparse(n,n);M1 = sparse(n,n);L = sparse(n,1);x = h:h:1; i=1;K(i,i)=(a((x(i)+x(i+1))/2)+a(x(i)/2))/h;i=n;K(i,i)=(a((x(i)+x(i+1))/2)+a((x(i)+x(i-1))/2))/h;for i=2:n-1 K(i,i)=(a((x(i)+x(i+1))/2)+a((x(i)+x(i-1))/2))/h;end for i=1:n-1 K(i,i+1) = -a( (x(i)+x(i+1))/2 )/h; K(i+1,i) = K(i,i+1);end

v = ones(n, 1) ;M1 = diag(4*v,0)+diag(v(1:n-1),-1)+diag(v(1:n-1),1);M = (h/6)*M1; A = K + M; L(1) = h*(f((x(1))/2)+f((x(1)+x(2))/2))/2;L(n) = h*(f((x(n)+x(n-1))/2)+f((x(n)+1)/2))/2;for i=2:n-1 L(i) = f((x(i)+x(i-1))/2)+f((x(i)+x(i+1))/2); L(i) = h*L(i)/2;endU = A\Luexact(h)Usol = [0;U;0];xnodes = [0;x'];ezplot('x*(1-x)',[0,1]);hold onplot(xnodes,Usol,'r-o')grid onhold off

Local Basis function

ix 1ix1ix

i

iIf we restrict any global basis functionon an interval Kj then it becomes eitherthe zero function or one of these local basis functions

1jxjx

Kj,1

1jxjx

Kj,2

1)( 1,1 jK xj

0)(,1 jK xj

21,1 )( cxcxjK

1211 cxc j

021 cxc j

jhc 1

1

j

j

h

xc 2

)()( 1,1 j

jhK xxxj

)()( 11

,2 jjhK xxx

j

1DFEM Implementation

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.0Partition of the interval [0,1]

]0.1 ,8.0 ,6.0 ,55.0 ,5.0 ,45.0 ,4.0 ,2.0 ,0[p Node Coordinate vector

Global labeling (nodes)

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.0

1 2 3 4 5 6 7 8 9

Element labeling (interval)1 2 3 4 5 6 7 8

Local labeling (nodes)

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.01 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

2 4 6 8

1 3 5 7

98765432

87654321tG

loba

l-loc

al

labe

ling

mat

rix

1st e

lem

2ed

elem

3ed

elem

8th e

lem

Global label of the first node

Global label of the second node

Example: Find the coordinate of the 2ed node in the fifth element

))5,2((tp

Three Matrices

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.0Partition of the interval [0,1]

]0.1 ,8.0 ,6.0 ,55.0 ,5.0 ,45.0 ,4.0 ,2.0 ,0[pNodes Coordinate vector

98765432

87654321tGlobal local labeling Matrix

Boundary nodes vector ]9,1[e

dxaa jijiji )''(),(

dxdxa jiji ''

A K MMass Matrixstiffness Matrix

Assemble the Mass Matrix

dxjijiM ,

Mass Matrix

9999939291

39333231

19232221

19131211

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

dxuu ),(

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.0n=9 # nodesM=8 # elements

xdxdxd 45.0

4.0

32

4.0

2.0

32

2.0

0

32

xd1

0

32

Contribution from K1

Contribution from K2

Contribution from K3

xdxdxd KKKK 45.0

4.0

3,1

4.0

2.0

2,22,1

2.0

0

1,2 00

xdxd 6.0

55.0

66

55.0

5.0

66 xd1

0

66

Contribution from K5

Contribution from K6

xdxd KKKK 6.0

55.0

6,26,2

55.0

5.0

5,15,1

Each entries of the matrix is a sum of several Contribution from elements. These contributions are integrals of two local basis functions in the same element

Element labeling (interval)1 2 3 4 5 6 7 8

1,2 K1,1 K

2,2 K2,1 K

8,2 K8,1 K

We generate all possible contributions in each element (this is called local mass matrix)

1

11

1

11

1

11

1

11

,2,2,1,2

,2,1,1,1

K

KK

K

KK

K

KK

K

KK

dxdx

dxdx

9

99

9

99

9

99

9

99

,2,2,1,2

,2,1,1,1

K

KK

K

KK

K

KK

K

KK

dxdx

dxdx

How many integrations ??

32 24

Assemble the Mass Matrix

Local Basis function

1jxjx

)()( ,11 xxx

jKjjh

)()( 11

,2 jjhK xxx

j

j

j

jj

j

j

jj

j

j

jj

j

j

jj

x

x

KK

x

x

KK

x

x

KK

x

x

KK

dxdx

dxdx

11

11

,2,2,1,2

,2,1,1,1

j

j

jj

x

x

KK dx1

,1,2

jh6

1

j

x

x

KK hdxj

j

jj 3

1,1,1

1

jh

x

x

KK

j

j

jjdx

3

1,2,2

1

21

12

6jh

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8 9

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.0

15

1

30

1

30

1

15

1

2.01 h

1 2

1

2

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

60

1

120

1

120

1

60

1

05.03 h

347

36

4

15/1

15/130/1

30/1

60/160/1120/1

120/1

15

1

30

1

30

1

15

1

2.02 h

21

32

2

3

15/1

15/130/1

30/1

60

1

120

1

120

1

60

1

05.03 h

457

46

5

60/160/1120/1

120/1

21

12

6jh

1DFEM Implementation

% main.m file

p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1];t=[1:8; 2:9];e=[1,9];[M] = createM(p,e,t)

21

12

6jh

% createM.m file function [M] = createM(p,e,t)n = length(p); % number of nodesm = size(t,2); % number of elementsM = sparse(n,n); for j=1:m loc2glb = t(1:2,j); % global labeling for the nodes in Kj xcoord = p(loc2glb); % coordinates of the nodes hj = xcoord(2)-xcoord(1); l_mass = (hj/6)*[2, 1; ... 1, 2]; % local mass matrix M(loc2glb,loc2glb) = M(loc2glb,loc2glb) + l_mass; % Add contributionsend

Stiffness Matrix

1jxjx

j

j

jj

j

j

jj

j

j

jj

j

j

jj

x

x

KK

x

x

KK

x

x

KK

x

x

KK

dxadxa

dxadxa

11

11

''''

''''

,2,2,1,2

,2,1,1,1

j

j

jj

x

x

KK dxa1

'' ,1,2

j

j

x

xjh

dxxa1

2 )(1)

2(

1,1,1

1

1

''

jj

j

j

j

jj

xxa

h

x

x

KK dxa

11

11

)2

(1 1jj

j

xxa

h

dxa ji ''jiK ,

Stiff Matrix

)2

(1 1

jj xxa

jh)

2(

1,2,2

1

1

''

jj

j

j

j

jj

xxa

h

x

x

KK dxa

)()( ,11 xxx

jKjjh

)()( 11

,2 jjhK xxx

j

% main.m file

p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1];t=[1:8; 2:9];e=[1,9];[K] = createK(p,e,t)

Stiffness Matrix

11

11)

2(

1 1jj

j

xxa

h

% createK.m filefunction [K] = createK(p,e,t)n = length(p); % number of nodesm = size(t,2); % number of elementsa = inline('x+2','x');K = sparse(n,n); for j=1:m loc2glb = t(1:2,j); % global labeling for the nodes in Kj xcoord = p(loc2glb); % coordinates of the nodes hj = xcoord(2)-xcoord(1); xb = (xcoord(1)+xcoord(2))/2; ab = a(xb); l_stif = (ab/hj)*[1, -1; ... -1, 1]; % local stiffness matrix K(loc2glb,loc2glb) = K(loc2glb,loc2glb) + l_stif; % Add contributionsend

Load vector

1jxjx

j

j

j

j

j

j

x

x

K

x

x

K

dxf

dxf

1

1

,2

,1

j

j

j

x

x

K dxf1

,1

dxf i

ibStiff Matrix

)2

(,1)2

(11 jj

j

jj xx

K

xx

j fh

)2

(2

1 jj xxjhf

j

j

j

x

x

K dxf1

,1

)2

(2

1 jj xxjhf

1

1)

2(

21jj xxjh

f

Local Load vector

)()( ,11 xxx

jKjjh

)()( 11

,2 jjhK xxx

j

% createL.m file

function [L] = createL(p,e,t)n = length(p); % number of nodesm = size(t,2); % number of elementsf = inline('-x^2+5*x+3','x');L = sparse(n,1); for j=1:m loc2glb = t(1:2,j); % global labeling for the nodes in Kj xcoord = p(loc2glb); % coordinates of the nodes hj = xcoord(2)-xcoord(1); xb = (xcoord(1)+xcoord(2))/2; fb = f(xb); l_L = (hj/2)*fb*[1; 1]; % local load vector L(loc2glb) = L(loc2glb) + l_L; % Add contributionsend

% main.m file

p=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1];t=[1:8; 2:9];e=[1,9];[L] = createL(p,e,t)

1

1)

2(

21jj xxjh

f

Load vector

0)1()0(

)''(

uu

fuau

Wea

k

)( )(),(

such that )( Find10

10

HLua

Hu

Boundary Conditions

}0)1()0(:)({)( 110 vvHvH

0)1(')0(

)''(

uu

fuau

Wea

k

HLua

Hu

)(),(

such that Find

}0)0(:)({ 1 vHvH)(1

0 HSh HSh

Example:1

0

Consider the partition

13

1

3

2},{span 21 hS

2

},,{span 321 hS

12 3

]4,1[e]1[e

1 2 3 4

Boundary Conditions

A K MMass Matrixstiffness Matrix

LAU

impose boundary conditions

5.0 55.0 6.0 8.0 0.12.0 4.0 45.00.0

1 2 3 4 5 6 7 8 9

0)0( u 0)1( u

We need to remove contributions coming from these two red local functions

for i=1:length(e) ib = e(i); A(ib,:)=sparse(1,n); A(:,ib)=sparse(n,1); A(ib,ib)=1; L(ib)=0;end

LAU

9

8

2

1

9

8

2

1

99

32

232221

1211

98

898887

78

0

0

L

L

L

L

U

U

U

U

aa

aaa

a

a

aaa

aa

1 0

0

10

0 0

0

Boundary Conditions% non uniform meshp=[0,0.2,0.4,0.45,0.5,0.55,0.6,0.8,1]; t=[1:8; 2:9];e=[1,9]; % % uniform mesh% p=0:0.125:1; n = length(p); t=[1:n-1;2:n];% e=[1,n]; n = length(p); % number of nodesm = size(t,2); % number of elementsuexact = inline('x*(1-x)','x'); [M] = createM(p,e,t)[K] = createK(p,e,t)[L] = createL(p,e,t)A = K + M; for i=1:length(e) ib = e(i); A(ib,:)=sparse(1,n); A(:,ib)=sparse(n,1); A(ib,ib)=1; L(ib)=0;endU = A\L;ezplot('x*(1-x)',[0,1]);hold on; plot(p,U,'r-o'); grid on; hold off

Impose boundary conditions

Uniform Mesh

Uniform mesh (using 8 elements)

Non-uniform mesh (using 8 elements)

Higher Order Polynomial

a more general finite element space, consisting of piecewise polynomials of degree r − 1, where r is an integer ≥ 2, with the above piecewise linear case included for r = 2, thus

010)( 1 )v(),v( on each K : vCv S jr-h

Πk denotes the space of polynomials of degree ≤ k.

Example:

r = 2 Sh space of cont pw-linear

Example: (HW)

r = 3 Sh space of cont pw-quadratic

1 2 3 4 5

322

1)( cxcxcx 322

1)( cxcxcx

Example: r = 4 Sh space of cont pw-cubic

432

23

1)( cxcxcxcx

1jx jxjj hx3

11

jj hx3

21

0)(

0)(

0)(

1)(

1

3

211

3

111

11

j

jj

jj

j

x

hx

hx

x

0)(

0)(

1)(

0)(

2

3

212

3

112

12

j

jj

jj

j

x

hx

hx

x

0)(

1)(

0)(

0)(

3

3

213

3

113

13

j

jj

jj

j

x

hx

hx

x

1)(

0)(

0)(

0)(

4

3

214

3

114

14

j

jj

jj

j

x

hx

hx

x

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

44434241

34333231

24232221

14131211

Local mass matrix

)','()','()','()','(

)','()','()','()','(

)','()','()','()','(

)','()','()','()','(

44434241

34333231

24232221

14131211

Local stiffness matrix

432

23

1)( cxcxcxcx

Higher Order Polynomial

1 2 3

1 2 3 4 5 6 7 8 9

****

****

****

****

1 2

1

2

10

3 4

3

4

****

****

****

****

4 5

4

5

6 7

6

7

****

****

****

****

7 8

7

8

910

93

10

Remark: The matrix is not tridiagonal

Theorem

r

rh vChvvI

0

r

rh vChvvI 1

1

)(for rHv

Approximation properties of Sh

r

rh uChuu 1

1

Theorem 5.1+5.2:

r

rh uChuu

0

)( rHu

432

23

1~~~~)(

~ctctctcx

1jx jxjj hx3

11

jj hx3

21

0)(

0)(

0)(

1)(

1

3

211

3

111

11

j

jj

jj

j

x

hx

hx

x

0)(

0)(

1)(

0)(

2

3

212

3

112

12

j

jj

jj

j

x

hx

hx

x

0)(

1)(

0)(

0)(

3

3

213

3

113

13

j

jj

jj

j

x

hx

hx

x

1)(

0)(

0)(

0)(

4

3

214

3

114

14

j

jj

jj

j

x

hx

hx

x

Reference element

1jx jxjj hx3

11

jj hx3

21

0 13

1

3

2

Reference element

mapping )( 11

jhxxt

j

0)1(~

0)(~

0)(~

1)0(~

1

3

21

3

11

1

0)1(~

0)(~

1)(~

0)0(~

2

3

22

3

12

2

0)1(~

1)(~

0)(~

0)0(~

3

3

23

3

13

3

1)1(~

0)(~

0)(~

0)0(~

4

3

24

3

14

4

0

0

0

1

~

~

~

~

1111

1

1

0000

4

3

2

1

3

2

9

4

27

83

1

9

1

27

1

c

c

c

c

0

0

0

1

~

~

~

~

0001

12

99

2

112

918

2

459

2

9

2

27

2

27

2

9

4

3

2

1

c

c

c

c

1

9

~

~

~

~

2

11

2

9

4

3

2

1

c

c

c

c

092

452

27

02

918

2

27

012

92

9

1

~2

~ 3

~ 4

~

Example: r = 4 Sh space of cont pw-cubic

432

23

1)( cxcxcxcx

1jx jx

0)('

0)(

0)('

1)(

1

1

11

11

j

j

j

j

x

x

x

x

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

44434241

34333231

24232221

14131211

Local mass matrix

)','()','()','()','(

)','()','()','()','(

)','()','()','()','(

)','()','()','()','(

44434241

34333231

24232221

14131211

Local stiffness matrix

432

23

1)( cxcxcxcx

C element1

0)('

0)(

1)('

0)(

2

2

12

12

j

j

j

j

x

x

x

x

0)('

1)(

0)('

0)(

3

3

13

13

j

j

j

j

x

x

x

x

1)('

0)(

0)('

0)(

4

4

14

14

j

j

j

j

x

x

x

x

Example:

432

23

1)( cxcxcxcx

1jx jx

0)('

0)(

0)('

1)(

1

1

11

11

j

j

j

j

x

x

x

x

C element .vs. C element1

0)('

0)(

1)('

0)(

2

2

12

12

j

j

j

j

x

x

x

x

0)('

1)(

0)('

0)(

3

3

13

13

j

j

j

j

x

x

x

x

1)('

0)(

0)('

0)(

4

4

14

14

j

j

j

j

x

x

x

x

432

23

1)( cxcxcxcx

1jx jxjj hx3

11

jj hx3

21

0)(

0)(

0)(

1)(

1

3

211

3

111

11

j

jj

jj

j

x

hx

hx

x

0)(

0)(

1)(

0)(

2

3

212

3

112

12

j

jj

jj

j

x

hx

hx

x

0)(

1)(

0)(

0)(

3

3

213

3

113

13

j

jj

jj

j

x

hx

hx

x

1)(

0)(

0)(

0)(

4

3

214

3

114

14

j

jj

jj

j

x

hx

hx

x

0 2

11

1C

0C

All global basis functions are continuous (only)

All global basis functions and their first derivatives are continuous

',

0)1()0(

''

uu

fu

Wea

k

)( )(),(

such that )( Find10

10

HLua

Hu

4th order differential equation

}0)1()0(:)({)( 110 vvHvH

0)1(')1()0(')0(

''''

uuuu

fu

Wea

k

)( )(),(

such that )( Find20

20

HLua

Hu

}0)1()0(:)({ 220 vvHvH

)(10 HSh

20HSh

1C

0C

It is required to use

It is required to use

element element